One intriguing and well known illusion is the Crazy Circle illusion in which the apparent circular motion of a series of dots (usually six) is revealed to really be only a rectilinear movement along fixed straight segments (You can check a good rendition here). Yet, this interpretation is biased by the usual appearance vs reality dichotomy that has plagued so much of our culture since Plato. In fact, a different interpretation is available and indeed preferable. I will show you that there is no illusion and that everything you see, both the circular motion and the rectilinear motion, is real. This is nothing new here, of course. Anyone who has studied should be aware of it. The catch is that movement is relative, so that the same object can have many different relative motions at the same time. In reality, as you will see, the dots are both moving along straight segments and circling around their center of mass. To make the point as clear as possible I have prepared a slightly different version of the illusion. Letís proceed step by step. First, take a look at the popular version of the moving dots.
You see them circling around a common center and, all together, also circling around another center. If we introduce the spokes, we soon notice that each dot is also moving along a straight line.
The rectilinear motion is more evident if only one or two dots at a time are shown.
The conclusion that the dots are not circling seems inescapable. Thus we resort to the notion of illusion: there is an apparent circular motion, but this is an illusion, since we suppose that really there are only rectilinear motions. Yet, such a conclusion is wrong. Let me explain why. Since Galileo, we have known that all motion is relative to a reference frame. There is no absolute motion (whatever Newton may have thought). Thus we can choose any reference frame we like Ė for instance, if we choose the center of mass of the dots as the reference frame, this is what we obtain.
As you can see, if we take the center of the circle of dots as the reference frame, unambiguous circular motion ensues. If you are not convinced, please put your fingertip on the little black circle on the screen and you'll see that the colored dots are circling around.
The conclusion is that there is not a real motion and an illusory motion; rather there are two relative motions (circular and rectilinear) and, because of the clever design, we are free to choose which one is perceptually more powerful. The main thing is that there is no illusion -- Each motion, linear and circular, is equally real and equally relative. We can repeat the process with only two dots and the structure will appear even more clearly.
Finally, we can play a bit with the mathematic parameters and obtain a series of equally amusing cases.